The Stacks project

83.15 The site associated to a semi-representable object

Let $\mathcal{C}$ be a site. Recall that a semi-representable object of $\mathcal{C}$ is simply a family $\{ U_ i\} _{i \in I}$ of objects of $\mathcal{C}$. A morphism $\{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ of semi-representable objects is given by a map $\alpha : I \to J$ and for every $i \in I$ a morphism $f_ i : U_ i \to V_{\alpha (i)}$ of $\mathcal{C}$. The category of semi-representable objects of $\mathcal{C}$ is denoted $\text{SR}(\mathcal{C})$. See Hypercoverings, Definition 25.2.1 and the enclosing section for more information.

For a semi-representable object $K = \{ U_ i\} _{i \in I}$ of $\mathcal{C}$ we let

\[ \mathcal{C}/K = \coprod \nolimits _{i \in I} \mathcal{C}/U_ i \]

be the disjoint union of the localizations of $\mathcal{C}$ at $U_ i$. There is a natural structure of a site on this category, with coverings inherited from the localizations $\mathcal{C}/U_ i$. The site $\mathcal{C}/K$ is called the localization of $\mathcal{C}$ at $K$. Observe that a sheaf on $\mathcal{C}/K$ is the same thing as a family of sheaves $\mathcal{F}_ i$ on $\mathcal{C}/U_ i$, i.e.,

\[ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \nolimits _{i \in I} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i) \]

This is occasionally useful to understand what is going on.

Let $\mathcal{C}$ be a site. Let $K = \{ U_ i\} _{i \in I}$ be an object of $\text{SR}(\mathcal{C})$. There is a continuous and cocontinuous localization functor $j : \mathcal{C}/K \to \mathcal{C}$ which is the product of the localization functors $j_ i : \mathcal{C}/V_ i \to \mathcal{C}$. We obtain functors $j_!$, $j^{-1}$, $j_*$ exactly as in Sites, Section 7.25. In terms of the product decomposition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \nolimits _{i \in I} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ we have

\[ \begin{matrix} j_! & : & (\mathcal{F}_ i)_{i \in I} & \longmapsto & \coprod j_{i, !}\mathcal{F}_ i \\ j^{-1} & : & \mathcal{G} & \longmapsto & (j_ i^{-1}\mathcal{G})_{i \in I} \\ j_* & : & (\mathcal{F}_ i)_{i \in I} & \longmapsto & \prod j_{i, *}\mathcal{F}_ i \end{matrix} \]

as the reader easily verifies.

Let $f : K \to L$ be a morphism of $\text{SR}(\mathcal{C})$. Then we obtain a continuous and cocontinuous functor

\[ v : \mathcal{C}/K \longrightarrow \mathcal{C}/L \]

by applying the construction of Sites, Lemma 7.25.8 to the components. More precisely, suppose $f = (\alpha , f_ i)$ where $K = \{ U_ i\} _{i \in I}$, $L = \{ V_ j\} _{j \in J}$, $\alpha : I \to J$, and $f_ i : U_ i \to V_{\alpha (i)}$. Then the functor $v$ maps the component $\mathcal{C}/U_ i$ into the component $\mathcal{C}/V_{\alpha (i)}$ via the construction of the aforementioned lemma. In particular we obtain a morphism

\[ f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L) \]

of topoi. In terms of the product decompositions $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \nolimits _{i \in I} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ and $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L) = \prod \nolimits _{j \in J} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V_ j)$ the reader verifies that

\[ \begin{matrix} f_! & : & (\mathcal{F}_ i)_{i \in I} & \longmapsto & (\coprod \nolimits _{i \in I, \alpha (i) = j} f_{i, !}\mathcal{F}_ i)_{j \in J} \\ f^{-1} & : & (\mathcal{G}_ j)_{j \in J} & \longmapsto & (f_ i^{-1}\mathcal{G}_{\alpha (i)})_{i \in I} \\ f_* & : & (\mathcal{F}_ i)_{i \in I} & \longmapsto & (\prod \nolimits _{i \in I, \alpha (i) = j} f_{i, *}\mathcal{F}_ i)_{j \in J} \end{matrix} \]

where $f_ i : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V_{\alpha (i)})$ is the morphism associated to the localization functor $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ corresponding to $f_ i : U_ i \to V_{\alpha (i)}$.

Lemma 83.15.1. Let $\mathcal{C}$ be a site.

  1. For $K$ in $\text{SR}(\mathcal{C})$ the functor $j : \mathcal{C}/K \to \mathcal{C}$ is continuous, cocontinuous, and has property P of Sites, Remark 7.20.5.

  2. For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $v : \mathcal{C}/K \to \mathcal{C}/L$ (see above) is continuous, cocontinuous, and has property P of Sites, Remark 7.20.5.

Proof. Proof of (2). In the notation of the discussion preceding the lemma, the localization functors $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ are continuous and cocontinuous by Sites, Section 7.25 and satisfy $P$ by Sites, Remark 7.25.11. It is formal to deduce $v$ is continuous and cocontinuous and has $P$. We omit the details. We also omit the proof of (1). $\square$

Lemma 83.15.2. Let $\mathcal{C}$ be a site and $K$ in $\text{SR}(\mathcal{C})$. For $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ we have

\[ j_*j^{-1}\mathcal{F} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (F(K)^\# , \mathcal{F}) \]

where $F$ is as in Hypercoverings, Definition 25.2.2.

Proof. Say $K = \{ U_ i\} _{i \in I}$. Using the description of the functors $j^{-1}$ and $j_*$ given above we see that

\[ j_*j^{-1}\mathcal{F} = \prod \nolimits _{i \in I} j_{i, *}(\mathcal{F}|_{\mathcal{C}/U_ i}) = \prod \nolimits _{i \in I} \mathop{\mathcal{H}\! \mathit{om}}\nolimits (h_{U_ i}^\# , \mathcal{F}) \]

The second equality by Sites, Lemma 7.26.3. Since $F(K) = \coprod h_{U_ i}$ in $\textit{PSh}(\mathcal{C}$, we have $F(K)^\# = \coprod h_{U_ i}^\# $ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and since $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (-, \mathcal{F})$ turns coproducts into products (immediate from the construction in Sites, Section 7.26), we conclude. $\square$

Lemma 83.15.3. Let $\mathcal{C}$ be a site.

  1. For $K$ in $\text{SR}(\mathcal{C})$ the functor $j_!$ gives an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\# $ where $F$ is as in Hypercoverings, Definition 25.2.2.

  2. The functor $j^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ corresponds via the identification of (1) with $\mathcal{F} \mapsto (\mathcal{F} \times F(K)^\# \to F(K)^\# )$.

  3. For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $f^{-1}$ corresponds via the identifications of (1) to the functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(L)^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\# $, $(\mathcal{G} \to F(L)^\# ) \mapsto (\mathcal{G} \times _{F(L)^\# } F(K)^\# \to F(K)^\# )$.

Proof. Observe that if $K = \{ U_ i\} _{i \in I}$ then the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ decomposes as the product of the categories $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$. Observe that $F(K)^\# = \coprod _{i \in I} h_{U_ i}^\# $ (coproduct in sheaves). Hence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\# $ is the product of the categories $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_{U_ i}^\# $. Thus (1) and (2) follow from the corresponding statements for each $i$, see Sites, Lemmas 7.25.4 and 7.25.7. Similarly, if $L = \{ V_ j\} _{j \in J}$ and $f$ is given by $\alpha : I \to J$ and $f_ i : U_ i \to V_{\alpha (i)}$, then we can apply Sites, Lemma 7.25.9 to each of the re-localization morphisms $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ to get (3). $\square$

Lemma 83.15.4. Let $\mathcal{C}$ be a site. For $K$ in $\text{SR}(\mathcal{C})$ the functor $j^{-1}$ sends injective abelian sheaves to injective abelian sheaves. Similarly, the functor $j^{-1}$ sends K-injective complexes of abelian sheaves to K-injective complexes of abelian sheaves.

Proof. The first statement is the natural generalization of Cohomology on Sites, Lemma 21.7.1 to semi-representable objects. In fact, it follows from this lemma by the product decomposition of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ and the description of the functor $j^{-1}$ given above. The second statement is the natural generalization of Cohomology on Sites, Lemma 21.20.1 and follows from it by the product decomposition of the topos.

Alternative: since $j$ induces a localization of topoi by Lemma 83.15.3 part (1) it also follows immediately from Cohomology on Sites, Lemmas 21.7.1 and 21.20.1 by enlarging the site; compare with the proof of Cohomology on Sites, Lemma 21.13.3 in the case of injective sheaves. $\square$

Remark 83.15.5 (Variant for over an object). Let $\mathcal{C}$ be a site. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The category $\text{SR}(\mathcal{C}, X)$ of semi-representable objects over $X$ is defined by the formula $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$. See Hypercoverings, Definition 25.2.1. Thus we may apply the above discussion to the site $\mathcal{C}/X$. Briefly, the constructions above give

  1. a site $\mathcal{C}/K$ for $K$ in $\text{SR}(\mathcal{C}, X)$,

  2. a decomposition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ if $K = \{ U_ i/X\} $,

  3. a localization functor $j : \mathcal{C}/K \to \mathcal{C}/X$,

  4. a morphism $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L)$ for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$.

All results of this section hold in this situation by replacing $\mathcal{C}$ everywhere by $\mathcal{C}/X$.

Remark 83.15.6 (Ringed variant). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$. In this case, for any semi-representable object $K$ of $\mathcal{C}$ the site $\mathcal{C}/K$ is a ringed site with sheaf of rings $\mathcal{O}_ K = j^{-1}\mathcal{O}_\mathcal {C}$. The constructions above give

  1. a ringed site $(\mathcal{C}/K, \mathcal{O}_ K)$ for $K$ in $\text{SR}(\mathcal{C})$,

  2. a decomposition $\textit{Mod}(\mathcal{O}_ K) = \prod \textit{Mod}(\mathcal{O}_{U_ i})$ if $K = \{ U_ i\} $,

  3. a localization morphism $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ of ringed topoi,

  4. a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L), \mathcal{O}_ L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C})$.

Many of the results above hold in this setting. For example, the functor $j^*$ has an exact left adjoint

\[ j_! : \textit{Mod}(\mathcal{O}_ K) \to \textit{Mod}(\mathcal{O}_\mathcal {C}), \]

which in terms of the product decomposition given in (2) sends $(\mathcal{F}_ i)_{i \in I}$ to $\bigoplus j_{i, !}\mathcal{F}_ i$. Similarly, given $f : K \to L$ as above, the functor $f^*$ has an exact left adjoint $f_! : \textit{Mod}(\mathcal{O}_ K) \to \textit{Mod}(\mathcal{O}_ L)$. Thus the functors $j^*$ and $f^*$ are exact, i.e., $j$ and $f$ are flat morphisms of ringed topoi (also follows from the equalities $\mathcal{O}_ K = j^{-1}\mathcal{O}_\mathcal {C}$ and $\mathcal{O}_ K = f^{-1}\mathcal{O}_ L$).

Remark 83.15.7 (Ringed variant over an object). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote $\mathcal{O}_ X = \mathcal{O}_\mathcal {C}|_{\mathcal{C}/U}$. Then we can combine the constructions given in Remarks 83.15.5 and 83.15.6 to get

  1. a ringed site $(\mathcal{C}/K, \mathcal{O}_ K)$ for $K$ in $\text{SR}(\mathcal{C}, X)$,

  2. a decomposition $\textit{Mod}(\mathcal{O}_ K) = \prod \textit{Mod}(\mathcal{O}_{U_ i})$ if $K = \{ U_ i\} $,

  3. a localization morphism $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X)$ of ringed topoi,

  4. a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L), \mathcal{O}_ L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$.

Of course all of the results mentioned in Remark 83.15.6 hold in this setting as well.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09WK. Beware of the difference between the letter 'O' and the digit '0'.